The Borsuk-Ulam property for homotopy classes of maps between the torus and the Klein bottle -- part 2

TL;DR

This paper characterizes which homotopy classes of maps from the 2-torus to the Klein bottle have the Borsuk-Ulam property under free involutions, using fundamental group homomorphisms, completing the analysis for this specific case.

Contribution

It determines the homotopy classes of maps from the 2-torus to the Klein bottle with the Borsuk-Ulam property for all free involutions of the torus, using algebraic topology methods.

Findings

Identifies homotopy classes with the Borsuk-Ulam property via fundamental group homomorphisms.

Provides a complete classification for maps from $T^2$ to $K^2$ under free involutions.

Uses algebraic invariants to characterize the property in this setting.

Abstract

Let $M$ be a topological space that admits a free involution $\tau$, and let $N$ be a topological space. A homotopy class $\beta \in [ M,N ]$ is said to have the Borsuk-Ulam property with respect to $\tau$ if for every representative map $f: M \to N$ of $\beta$, there exists a point $x \in M$ such that $f(\tau(x))= f(x)$. In this paper, we determine the homotopy class of maps from the $2$-torus $T^2$ to the Klein bottle $K^2$ that possess the Borsuk-Ulam property with respect to any free involution of $T^2$ for which the orbit space is $K^2$. Our results are given in terms of a certain family of homomorphisms involving the fundamental groups of $T^2$ and $K^2$. This completes the analysis of the Borsuk-Ulam problem for the case $M=T^2$ and $N=K^2$, and for any free involution $\tau$ of $T^2$.